Zoran: there are several things called “Birkhoff’s theorem” in various field of mathematics and mathematical physics, and belong even to at least 2 different classical Birkhoff’s. Even wikipedia has pages for more than one such theorem. To me the first which comes to mind is Birkhoff’s factorization theorem, now also popular in Kreimer-Connes-Marcolli work and in connection to loop groups (cf. book by Segal nad Pressley). I would like that the nnlab does not mislead by distinguishing one of the several famous Bikhoff labels without mentioning and directing to 2-3 others.

Ian Durham: Good point. I think this probably ought to be renamed the “Birkhoff-von Neumann theorem.” Is that a good enough label or should we get more specific with it?

Toby: I have moved it. See also the new page Birkhoff's theorem, which is basically just Zoran's comment above.

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The Birkhoff-von Neumann theorem

Given a permutation σ∈Sn\sigma \in S_n, the permutation matrix that is associated with σ\sigma is the n×nn \times n matrix Pσ(j,k)P_{\sigma}(j,k), where 1≤j,k≤n1 \le j,k \le n, whose entries are given by

An n×nn \times n doubly stochastic matrix DD is a square matrix whose elements are real and whose rows and columns sum to unity. Given such a matrix, there exist nonnegative weights {wσ:σ∈Sn}\{w_{\sigma}:\sigma \in S_n\} such that

The set of such matrices of order nn is said to form the convex hull of permutation matrices of the same order where the latter are the vertices (extreme points) of the former.

Quantum extensions

In the quantum context, doubly stochastic matrices become doubly stochastic channels, i.e. completely positive maps preserving both the trace and the identity. Quantum mechanically we understand the permutations to be the unitarily implemented channels. That is, we expect doubly stochastic quantum channels to be convex combinations of unitary channels (that is channels that can be represented by some combination of unitary transformations). Unfortunately it is well-known that some quantum channels cannotcannot be written that way.

However, large tensor powers of a channel may be easier to represent in this way, because one need not use only product unitaries in the decomposition. Thus one proposed solution to the problem is to denote, for any doubly stochastic channel TT, by dB(T)d_{B}(T) the Birkhoff defect, defined as the cb-norm distance from TT to the convex hull of the unitarily implemented channels. Then the key is to determine whether dB(T⊗n)d_{B}(T^{\otimes n}) goes to zero as n→∞n\to\infty.